Question: Simplify the following expression: $a = \dfrac{70z^3 + 49z^2}{-7z^3 + 35z^2}$ You can assume $z \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $70z^3 + 49z^2 = (2\cdot5\cdot7 \cdot z \cdot z \cdot z) + (7\cdot7 \cdot z \cdot z)$ The denominator can be factored: $-7z^3 + 35z^2 = - (7 \cdot z \cdot z \cdot z) + (5\cdot7 \cdot z \cdot z)$ The greatest common factor of all the terms is $7z^2$ Factoring out $7z^2$ gives us: $a = \dfrac{(7z^2)(10z + 7)}{(7z^2)(-z + 5)}$ Dividing both the numerator and denominator by $7z^2$ gives: $a = \dfrac{10z + 7}{-z + 5}$